Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2008, Article ID 570825, 23 pages doi:10.1155/2008/570825 Research Article Peristaltic Flow of a Magneto-Micropolar Fluid: Effect of Induced Magnetic Field Kh. S. Mekheimer Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt Correspondence should be addressed to Kh. S. Mekheimer, kh [email protected]Received 2 June 2008; Accepted 29 September 2008 Recommended by Jacek Rokicki We carry out the effect of the induced magnetic field on peristaltic transport of an incompressible conducting micropolar fluid in a symmetric channel. The flow analysis has been developed for low Reynolds number and long wavelength approximation. Exact solutions have been established for the axial velocity, microrotation component, stream function, magnetic-force function, axial- induced magnetic field, and current distribution across the channel. Expressions for the shear stresses are also obtained. The effects of pertinent parameters on the pressure rise per wavelength are investigated by means of numerical integrations, also we study the effect of these parameters on the axial pressure gradient, axial-induced magnetic field, as well as current distribution across the channel and the nonsymmetric shear stresses. The phenomena of trapping and magnetic-force lines are further discussed. Copyright q 2008 Kh. S. Mekheimer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction It is well known that many physiological fluids behave in general like suspensions of deformable or rigid particles in a Newtonian fluid. Blood, for example, is a suspension of red cells, white cells, and platelets in plasma. Another example is cervical mucus, which is a suspension of macromolecules in a water-like liquid. In view of this, some researchers have tried to account for the suspension behavior of biofluids by considering them to be non- Newtonian 1–6. Eringen 7introduced the concept of simple microfluids to characterise concentrated suspensions of neutrally buoyant deformable particles in a viscous fluid where the individuality of substructures affects the physical outcome of the flow. Such fluid models can be used to rheologically describe polymeric suspensions, normal human blood, and so forth, and have found applications in physiological and engineering problems 8–10. A subclass of these microfluids is known as micropolar fluids where the fluid microelements are considered
Transcript
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2008, Article ID 570825, 23 pagesdoi:10.1155/2008/570825
Research ArticlePeristaltic Flow of a Magneto-Micropolar Fluid:Effect of Induced Magnetic Field
Kh. S. Mekheimer
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt
Correspondence should be addressed to Kh. S. Mekheimer, kh [email protected]
Received 2 June 2008; Accepted 29 September 2008
Recommended by Jacek Rokicki
We carry out the effect of the induced magnetic field on peristaltic transport of an incompressibleconducting micropolar fluid in a symmetric channel. The flow analysis has been developed forlow Reynolds number and long wavelength approximation. Exact solutions have been establishedfor the axial velocity, microrotation component, stream function, magnetic-force function, axial-induced magnetic field, and current distribution across the channel. Expressions for the shearstresses are also obtained. The effects of pertinent parameters on the pressure rise per wavelengthare investigated by means of numerical integrations, also we study the effect of these parameterson the axial pressure gradient, axial-induced magnetic field, as well as current distribution acrossthe channel and the nonsymmetric shear stresses. The phenomena of trapping and magnetic-forcelines are further discussed.
Copyright q 2008 Kh. S. Mekheimer. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
1. Introduction
It is well known that many physiological fluids behave in general like suspensions ofdeformable or rigid particles in a Newtonian fluid. Blood, for example, is a suspension ofred cells, white cells, and platelets in plasma. Another example is cervical mucus, which is asuspension of macromolecules in a water-like liquid. In view of this, some researchers havetried to account for the suspension behavior of biofluids by considering them to be non-Newtonian [1–6].
Eringen [7] introduced the concept of simple microfluids to characterise concentratedsuspensions of neutrally buoyant deformable particles in a viscous fluid where theindividuality of substructures affects the physical outcome of the flow. Such fluid models canbe used to rheologically describe polymeric suspensions, normal human blood, and so forth,and have found applications in physiological and engineering problems [8–10]. A subclass ofthese microfluids is known as micropolar fluids where the fluid microelements are considered
2 Journal of Applied Mathematics
to be rigid [11, 12]. Basically, these fluids can support couple stresses and body couples andexhibit microrotational and microinertial effects.
The phenomenon of peristalsis is defined as expansion and contraction of an extensibletube in a fluid generate progressive waves which propagate along the length of the tube,mixing and transporting the fluid in the direction of wave propagation. It is an inherentproperty of many tubular organs of the human body. In some biomedical instruments, suchas heart-lung machines, peristaltic motion is used to pump blood and other biological fluids.It plays an indispensable role in transporting many physiological fluids in the body in varioussituations such as urine transport from the kidney to the bladder through the ureter, transportof spermatozoa in the ductus efferentes of the male reproductive tract, movement of ovumin the fallopian tubes, vasomotion of small blood vessels, as well as mixing and transportingthe contents of the gastrointestinal passage.
Peristaltic pumping mechanisms have been utilized for the transport of slurries,sensitive or corrosive fluids, sanitary fluid, noxious fluids in the nuclear industry, andmany others. In some cases, the transport of fluids is possible without moving internalmechanical components as in the case with peristaltically operated microelectromechanicalsystem devices [13].
The study of peristalsis in the context of fluid mechanics has received considerableattention in the last three decades, mainly because of its relevance to biological systemsand industrial applications. Several studies have been made, especially for the peristalsis innon-Newtonian fluids which have promising applications in physiology [14–23]. The mainadvantage of using a micropolar fluid model to study the peristaltic flow of suspensions incomparison with other classes of non-Newtonian fluids is that it takes care of the rotation offluid particles by means of an independent kinematic vector called the microrotation vector.
Magnetohydrodynamic (MHD) is the science which deals with the motion of a highlyconducting fluids in the presence of a magnetic field. The motion of the conducting fluidacross the magnetic field generates electric currents which change the magnetic field, and theaction of the magnetic field on these currents gives rise to mechanical forces which modify theflow of the fluid [24]. MHD flow of a fluid in a channel with elastic, rhythmically contractingwalls (peristaltic flow) is of interest in connection with certain problems of the movementof conductive physiological fluids (e.g., the blood and blood pump machines) and with theneed for theoretical research on the operation of a peristaltic MHD compressor. Effect of amoving magnetic field on blood flow was studied by Stud et al. [25], Srivastava and Agrawal[26] considered the blood as an electrically conducting fluid and it constitutes a suspension ofred cells in plasma. Also Agrawal and Anwaruddin [27] studied the effect of magnetic fieldon blood flow by taking a simple mathematical model for blood through an equally branchedchannel with flexible walls executing peristaltic waves using long wavelength approximationmethod.
Some recent studies [28–41] have considered the effect of a magnetic field on peristalticflow of a Newtonian and non-Newtonian fluids, and in all of these studies the effect of theinduced magnetic field have been neglected.
The first investigation of the effect of the induced magnetic field on peristaltic flowwas studied by Vishnyakov and Pavlov [42] where they considers the peristaltic MHD flowof a conductive Newtonian fluid; they used the asymptotic narrow-band method to solve theproblem and only obtained the velocity profiles in certain channel cross-sections for definiteparameter values. Currently, there is only two attempts [43, 44] for a study of the effectof induced magnetic field, one for a couple-stress fluid and the other for a non-Newtonianfluid (biviscosity fluid). To the best of our knowledge , the influence of a magnetic field on
Kh. S. Mekheimer 3
peristaltic flow of a conductive micropolar fluid has not been investigate with or without theinduced magnetic field.
With keeping the above discussion in mind, the goal of this investigation is to studythe effect of the induced magnetic field on peristaltic flow of a micropolar fluid (as a bloodmodel). The flow analysis is developed in a wave frame of reference moving with thevelocity of the wave. The problem is first modeled and then solved analytically for thestream function, magnetic-force function, and the axial pressure gradient. The results forthe pressure rise , shear stresses, the axial induced magnetic field, and the distribution ofthe current density across the channel have been discussed for various values of the problemparameters. Also, the contour plots for the magnetic force and stream functions are presented,the pumping characteristics and the trapping phenomena are discussed in detail. Finally, Themain conclusions are summarized in the last section.
2. Mathematical modelling
Consider the unsteady hydromagnetic flow of a viscous, incompressible, and electricallyconducting micropolar fluid through an axisymmetric two-dimensional channel of uniformthickness with a sinusoidal wave traveling down its wall. We choose a rectangularcoordinate system for the channel with X′ along the centerline in the direction of wavepropagation and Y ′ transverse to it. The system is stressed by an external transverse uniformconstant magnetic field of strength H ′0, which will give rise to an induced magnetic fieldH ′(h′X′(X
′, Y ′, t′), h′Y ′(X′, Y ′, t′), 0) and the total magnetic field will be H ′+(h′X′(X
′, Y ′, t′),H ′0 +h′Y ′(X
′, Y ′, t′), 0). The plates of the channel are assumed to be nonconductive, and thegeometry of the wall surface is defined as
h′(X′, t′) = a + b cos2πλ
(X′ − c t′), (2.1)
where a0 is the half-width at the inlet, b is the wave amplitude, λ is the wavelength, c is thepropagation velocity, and t′ is the time.
Neglecting the body couples, the equations of motion for unsteady flow of anincompressible micropolar fluid are
�∇· �V ′ = 0,
ρ
{∂�V ′
∂t′+ �V ′·�∇ �V ′
}= −�∇p′ + k�∇ × �w′ + (μ + k)�∇2 �V ′ + ρ �f ′,
ρj
{∂�V ′
∂t′+ �V ′·�∇ �w′
}= −2k �w′ + k�∇ × �V ′ − γ
(�∇ × �∇ × �w′
)+ (α + β + γ)∇
(�∇· �w′
),
(2.2)
where �V ′ is the velocity vector, �w′ is the microrotation vector, p′ is the fluid pressure, �f ′
is the body force, and ρ and j are the fluid density and microgyration parameter. Further,the material constants (new viscosities of the micropolar fluid) μ, k, α, β, and γ satisfy thefollowing inequalities (obtained by Eringen [11] ):
2μ + k ≥ 0, k ≥ 0, 3α + β + γ ≥ 0, γ ≥ |β|. (2.3)
4 Journal of Applied Mathematics
The governing equations for a magneto-micropolar fluid are as follows:Maxwell’s equations
∇· �H ′ = 0, ∇·�E′ = 0, (2.4)
∇ ∧ �H ′ = �J ′, with �J ′ = σ{�E′ + μe
(�V ′ ∧ �H ′+
)}, (2.5)
∇ ∧ �E′ = −μe∂ �H ′
∂t′, (2.6)
the continuity equation
∇· �V ′ = 0, (2.7)
the equations of motion
ρ
{∂ �V ′
∂ t′+(�V ′·�∇
)�V ′}
= −∇(p′ +
12μe
(H ′+
)2)+ (μ + k)∇2 �V ′ + k�∇ × �w′ − μe
(�H ′+·�∇
)�H ′+,
ρj
{∂�V ′
∂t′+(�V ′·�∇
)�w′}
= −2k �w′ + k�∇ × �V ′ − γ(�∇ × �∇ × �w′
)+ (α + β + γ)�∇
(�∇· �w′
),
∇2 =∂2
∂X′2+
∂2
∂Y ′2,
(2.8)
where �E′ is an induced electric field, �J ′ is the electric current density, μe is the magneticpermeability, and σ is the electrical conductivity.
Combining (2.4) and (2.5)–(2.7), we obtain the induction equation:
∂ �H ′+
∂t′= �∇ ∧
{�V ′ ∧ �H ′+
}+
1ζ∇2 �H ′+, (2.9)
where ζ = 1/σμe is the magnetic diffusivity.We should carry out this investigation in a coordinate system moving with the wave
speed c, in which the boundary shape is stationary. The coordinates and velocities in thelaboratory frame (X′, Y ′) and the wave frame (x′, y′) are related by
x′ = X′ − ct′, y′ = Y ′,
u′ = U′ − c, v′ = V ′,(2.10)
where U′, V ′, and u′, v′ are the velocity components in the corresponding coordinatesystems.
Kh. S. Mekheimer 5
Using these transformations and introducing the dimensionless variables
x =x′
λ, y =
y′
a, u =
u′
c, v =
λv′
ac, h =
h′(x′)a
,
p =a2
λμcp′(x′), t =
ct′
λ, j =
j ′
a2, ψ =
ψ ′
ca, φ =
φ′
H0a,
(2.11)
we find that the equations which govern the MHD flow for a micropolar fluid in terms of thestream function ψ(x, y) and magnetic-force function φ(x, y) are
Reδ{(
ψy∂
∂x− ψx
∂
∂y
)ψy
}= −
∂pm∂x
+1
1 −N∇2ψy +
N
1 −N∂w
∂y+ ReS2φyy
+ ReS2δ
(φy
∂
∂x− φx
∂
∂y
)φy,
(2.12)
Reδ3{(
ψx∂
∂y− ψy
∂
∂x
)ψx
}= −
∂pm∂y− δ2
1 −N∇2ψx −
δ2N
1 −N∂w
∂x− ReS2δ2φxy
− ReS2δ3(φy
∂
∂x− φx
∂
∂y
)φx,
(2.13)
Reδj(
1 −NN
){(ψy
∂
∂x− ψx
∂
∂y
)w
}= −2w − ∇2ψ +
(2 −Nm2
)∇2w, (2.14)
ψy − δ(ψyφx − ψxφy) +1Rm∇2φ = E, (2.15)
where
u =∂ψ
∂y, v = −δ
∂ψ
∂x, hx =
∂φ
∂y, hy = −δ
∂φ
∂x,
∇2 = δ2 ∂2
∂x2+
∂2
∂y2,
(2.16)
and the dimensionless parameters as follows:
(i) Reynolds number Re = caρ/μ,
(ii) wave number δ = a/λ,
(iii) Strommer’s number (magnetic-force number) S = (H0/c)√μe/ρ,
(iv) the magnetic Reynolds number Rm = σμeac,
(v) the coupling number N = k/(k + μ) (0 ≤ N ≤ 1), m2 = a2k(2μ + k)/(γ(μ + k)) isthe micropolar parameter,
(vi) the total pressure in the fluid, which equals the sum of the ordinary and magneticpressure, is pm = p + (1/2)Reδ(μe(H+)2/ρc2), and E(= −E/cH0) is the electric fieldstrength. The parameters α, β do not appear in the governing equations as themicrorotation vector is solenoidal. However, (2.12)–(2.15) reduce to the classicalMHD Navier-Stokes equations as k → 0.
6 Journal of Applied Mathematics
Excluding the total pressure from (2.12) and (2.13), we obtain
Reδ{(
ψy∂
∂x− ψx
∂
∂y
)∇2ψ
}=
11 −N∇
4ψ +N
1 −N∇2w + ReS2∇2φy
+ ReS2δ
(φy
∂
∂x− φx
∂
∂y
)∇2φ,
Reδj(
1 −NN
){(ψy
∂
∂x− ψx
∂
∂y
)w
}= −2w − ∇2ψ +
(2 −Nm2
)∇2w.
(2.17)
The instantaneous volume flow rate in the fixed frame is given by
Q =∫h′
0U′
(X′, Y ′, t
)dY ′, (2.18)
where h′ is a function of X′ and t.The rate of volume flow in the wave frame is given by
q =∫h′
0u′(x′, y′
)dy′, (2.19)
where h′ is a function of x′ alone. If we substitute (2.10) into (2.18) and make use of (2.19),we find that the two rates of volume flow are related through
Q = q + ch′. (2.20)
The time mean flow over a period T at a fixed position X′ is defined as:
Q =1T
∫T
0Qdt. (2.21)
Substituting (2.20) into (2.21), and integrating, we get
Q = q + ac. (2.22)
On defining the dimensionless time-mean flows θ and F, respectively, in the fixed andwave frame as
θ =Q
ac, F =
q
ac, (2.23)
one finds that (2.22) may be written as
θ = F + 1, (2.24)
Kh. S. Mekheimer 7
where
F =∫h
0
∂ψ
∂ydy = ψ(h) − ψ(0). (2.25)
We note that h represents the dimensionless form of the surface of the peristaltic wall:
h(x) = 1 + α cos(2πx), (2.26)
where
α =b
a(2.27)
is the amplitude ratio or the occlusion.If we select the zero value of the streamline at the streamline (y = 0)
ψ(0) = 0, (2.28)
then the wall (y = h) is a streamline of value
ψ(h) = F. (2.29)
For a non-conductive elastic channel wall, the boundary conditions for the dimen-sionless stream function ψ(x, y) and magnetic-force function φ(x, y) in the wave frame are[42, 44]
ψ = 0,∂2ψ
∂y2= 0, w = 0,
∂φ
∂y= 0 on y = 0,
∂ψ
∂y= −1, ψ = F, w = 0, φ = 0,
∂φ
∂y= 0 on y = h(x).
(2.30)
Under the long wavelength and low Reynolds number consideration [4–6, 34–36], thedimensionless equations of the problem are expressed in the following form:
∂4ψ
∂y4+N
∂2w
∂y2+ ReS2(1 −N)
∂3φ
∂y3= 0,
2 −Nm2
∂2w
∂y2= 2w +
∂2ψ
∂y2, (2.31)
∂2φ
∂y2= Rm
(E −
∂ψ
∂y
). (2.32)
8 Journal of Applied Mathematics
Combining these equations gives
ψ =1
H2(1 −N)
{2 −Nm2
(∂2w
∂y2−m2w
)+ ηy − C1y − C2
},
∂4w
∂y4−{m2 +H2(1 −N)
}∂2w
∂y2+
2m2H2(1 −N)2 −N w = 0,
(2.33)
where H2 = ReS2Rm, H = (μeH0)a√σ/μ is the Hartmann number (suitably greater than√
2), η = EH2(1 −N), and C1, C2 are an integration constants.
3. Exact solution
The general solutions of the microrotation component w and the stream function ψ are
w = Acosh(θ1y) + B sinh(θ1y) + Ccosh(θ2y) +D sinh(θ2y),
ψ =1
H2(1 −N)
{2 −Nm2
[(θ21 −m
2)(Acosh(θ1y) + B sinh(θ1y))
+ (θ22 −m
2)(Ccosh(θ2y) +D sinh(θ2y))] + ηy − C1y − C2
},
(3.1)
where
θ1 =1√2
√√√√((1 −N)H2 +m2) +
√((1 −N)H2 +m2)2 − 4
(2m2(1 −N)H2
(2 −N)
),
θ2 =1√2
√√√√((1 −N)H2 +m2) −
√((1 −N)H2 +m2)2 − 4
(2m2(1 −N)H2
(2 −N)
).
(3.2)
Using the corresponding boundary conditions in (2.20), we get
Thus the stream function and the microrotation component w will take the forms
ψ(x, y) =(F + h)
ζ
{ξ2
sinh(θ2y)sinh(θ2h)
− ξ1sinh(θ1y)sinh(θ1h)
}
−{
1 +(F + h)
ζ(θ2ξ2coth(θ2h) − θ1ξ1coth(θ1h))y
},
w(x, y) =(F + h)(1 −N)H2m2
(2 −N)ζ
{sinh(θ2y)sinh(θ2h)
−sinh(θ1y)sinh(θ1h)
}.
(3.4)
Now solving (2.32) with the corresponding boundary conditions in (2.30), we get themagnetic force function in the form
φ(x, y) = Rm
{ξ1(F + h)
θ1ζ
cosh(θ1y)sinh(θ1h)
− ξ2(F + h)θ2ζ
cosh(θ2y)sinh(θ2h)
+y2
2
(1 +
F + hζ
(θ2ξ2coth(θ2h) − θ1ξ1coth(θ1h)) + E)}
+ C3y + C4,
(3.5)
where
C3 = 0,
C4 = −Rm
{ξ1(F + h)
θ1ζ
cosh(θ1h)sinh(θ1h)
− ξ2(F + h)θ2ζ
cosh(θ2h)sinh(θ2h)
− h2
2
(1 +
(F + h)ζ
(θ2ξ2coth(θ2h) − θ1ξ1coth(θ1h)
)+ E
)}.
(3.6)
Also, the axial-induced magnetic field and the current density distribution across thechannel will take the forms
hx(x, y) = Rm
{ξ1(F + h)
ζ
sinh(θ1y)sinh(θ1y)
− ξ2(F + h)ζ
sinh(θ2y)sinh(θ2h)
+ y(
1 +F + hζ
(θ2ξ2coth(θ2h) − θ1ξ1coth(θ1h)
)+ E
)},
Jz(x, y) = Rm
{θ1ξ1(F + h)
ζ
cosh(θ1y)sinh(θ1h)
− θ2ξ2(F + h)ζ
cosh(θ2y)sinh(θ2h)
+(
1 +F + hζ
(θ2ξ2coth(θ2h) − θ1ξ1coth(θ1h)
)+ E
)}.
(3.7)
In the formulation under consideration, the field strength E is the determining factorand its value can be found by integrating (2.32), which represents Ohm’s law in differentialform, across the channel, taking into account the boundary conditions for φ and ψ in (2.30).In this case, we obtain the dimensionless E = F/h.
10 Journal of Applied Mathematics
When the flow is steady in the wave frame, one can characterize the pumpingperformance by means of the the pressure rise per wavelength. So, the axial pressure gradientcan be obtained from the equation
∂p
∂x=
1(1 −N
∂3ψ
∂y3+
N
1 −N∂w
∂y+H2
(E −
∂ψ
∂y
). (3.8)
Using (3.4), the axial pressure gradient will take the form
∂p
∂x=
(F + h)θ2
ζ
{ξ2
(θ2
2
1 −N −H2)+NH2m2
2 −N
}cosh(θ2y)sinh(θ2h)
− (F + h)θ1
ζ
{ξ1
(θ2
1
1 −N −H2)+NH2m2
2 −N
}cosh(θ1y)sinh(θ1h)
+H2{(E + 1) − (F + h)
ζ(θ1ξ1coth(θ1h) − θ2ξ2coth(θ2h))
}.
(3.9)
The pressure rise Δpλ for a channel of length L in its nondimensional forms is givenby
Δpλ =∫1
0
∂p
∂xdx. (3.10)
The integral in (3.10) is not integrable in closed form, it is evaluated numerically using adigital computer.
An interesting property of the micropolar fluid is that the stress tensor is notsymmetric. The nondimensional shear stresses in the problem under consideration are givenby
τxy =∂2ψ
∂y2− N
1 −Nw,
τyx =(
11 −N
)∂2ψ
∂y2+
N
1 −Nw.
(3.11)
The shear stresses τxy and τyx are calculated at both the lower and upper walls andgraphical results are shown in Figures 4–6.
4. Numerical results and discussion
This section is divided into three subsections. In the first subsection, the effects of variousparameters on the pumping characteristics of a magneto-miropolar fluid are investigated.The magnetic field characteristics are discussed in the second subsection. The trappingphenomenon and the magnetic-force lines are illustrated in the last subsection.
Kh. S. Mekheimer 11
0 0.2 0.4 0.6 0.8 1
x
0
5
10
15
20
25
30
35
dp/dx
m = 0.001m = 10m = 100N = 0.2
N = 0.4N = 0.6Newtonian
Figure 1: The axial pressure gradient versus the wavelength for α = 0.3, θ = −1.2, H = 2 and differentvalues of m and N.
−1 −0.5 0 0.5 1
θ
−15
−10
−5
0
5
10
15
20
Δpλ
H = 8
H = 2
IV I
III II
m = 0.001m = 10m = 100Newtonian
m = 0.001m = 10m = 10
Figure 2: The pressure rise versus flow rate for α = 0.4, N = 0.4, H = 2, and H = 8 at different values ofm.
4.1. Pumping characteristics
This subsection describes the influences of various emerging parameters of our analysison the axial pressure gradient ∂p/∂x, the pressure rise per wavelength Δpλ, and the shearstresses τxy, τyx on the lower and upper walls. The effects of these parameters are shown inFigures 1–6, and in most of the figures, the case of N → 0 corresponds to that of Newtonianfluid.
Figure 1 illustrates the variation of the axial pressure gradient with x for differentvalues of the microrotation parameter m and the coupling number N. We can see that in the
12 Journal of Applied Mathematics
−1 −0.5 0 0.5 1
θ
−20
−15
−10
−5
0
5
10
15
20
25
Δpλ
H = 8
H = 2
N = 0.2N = 0.4N = 0.6Newtonian
N = 0.2N = 0.4N = 0.4
Figure 3: The pressure rise versus flow rate for α = 0.4, m = 2, H = 2, and H = 8 at different values of N.
−1 −0.5 0 0.5 1
x
−10
−5
0
5
10
τ xy
Lower wall
Upper wall
H = 2H = 4H = 6
Figure 4: The shear stresses τxy for α = 0.5, θ = 1.2, m = 3, N = 0.6, and different values of H.
wider part of the channel xε [0, 0.2] and [0.8,1.0], the pressure gradient is relatively small,that is, the flow can easily pass without imposition of large pressure gradient. Where, ina narrow part of the channel xε [0.2, 0.8], a much larger pressure gradient is required tomaintain the same flux to pass it, especially for the narrowest position near x = 0.5. This is inwell agreement with the physical situation. Also from this figure, we observe the effect of mandN on the pressure gradient for fixed values of the other parameters, where the amplitudeof dp/dx decrease as m increases and increases with increasing N, and the smallest valueof such amplitude corresponds to the case N → 0 (Newtonian fluid). The effect of theHartmann number H on dp/dx is not included, where it is illustrated in a previous paper[44], where the amplitude of dp/dx increases as H increases.
Kh. S. Mekheimer 13
−1 −0.5 0 0.5 1
x
−15
−10
−5
0
5
10
15
τ yx
Lower wall
Upper wall
H = 2H = 4H = 6
Figure 5: The shear stresses τyx for α = 0.5, θ = 1.2, m = 3, N = 0.6, and different values of H.
−1 −0.5 0 0.5 1
x
−45
−40
−35
−30
−25
−20
−15
−10
−5
τ yx
NewtonianN = 0.2, m = 2N = 0.4N = 0.8
m = 2, N = 0.7m = 8m = 20
Figure 6: The shear stresses τyx for α = 0.5, θ = 1.2, H = 3, and different values of m and N.
Figures 2 and 3 illustrate the change of the pressure rise Δpλ versus the time-averagedmean flow rate θ for various values of the parametersm (= 0.001, 10,100 withN = 0.4, α = 0.4and different values of H) and N(= 0.2, 0.4, 0.6 with m = 2, α = 0.4 and different values ofH).
The graph is sectored so that the upper right-hand quadrant (I) denotes the regionof peristaltic pumping, where θ > 0 (positive pumping) and Δpλ > 0 (adverse pressuregradient). Quadrant (II), where Δpλ < 0 (favorable pressure gradient) and θ > 0 (positivepumping), is designated as augmented flow (copumping region). Quadrant (IV), such thatΔpλ > 0 (adverse pressure gradient) and θ < 0, is called retrograde or backward pumping.The flow is opposite to the direction of the peristaltic motion, and there is no flows in the
14 Journal of Applied Mathematics
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
hx
−1
−0.5
0
0.5
1
y
H = 8
H = 2
m = 5m = 20m = 100Newtonian
m = 5m = 20m = 100
Figure 7: Variation of the axial-induced magnetic field across the channel for α = 0.4, θ = 1.2, N =0.9, Rm = 1, and different values of m and H.
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
hx
−1.5
−1
−0.5
0
0.5
1
1.5
y
H = 8
H = 2
N = 0.2N = 0.4N = 0.6Newtonian
N = 0.2N = 0.4N = 0.6
Figure 8: Variation of the axial-induced magnetic field across the channel for α = 0.4, θ = 1.2, m = 2, Rm =1, and different values of N and H.
last quadrant (Quadrant (III)). It is shown in both Figures 2 and 3, that there is an inverselylinear relation between Δpλ and θ, that is, the pressure rise decreases with increasing the flowrate and the pumping curves are linear both for Newtonian and micropolar fluid. Moreover,the pumping curves for micropolar fluid lie above the Newtonian fluid in pumping region(Δpλ > 0), but asm increases, the curves tend to coincide. In copumping region (Δpλ < 0), thepumping increases with an increase in m. Figure 3 shows the effects of the coupling numberN on Δpλ, where the pumping increases with an increase in N and the pumping curve for
Kh. S. Mekheimer 15
−0.4 −0.2 0 0.2 0.4
hx
−1.5
−1
−0.5
0
0.5
1
1.5
y
α = 0α = 0.3
α = 0.6α = 0.8
Figure 9: Variation of the axial-induced magnetic field across the channel for m = 3, N = 0.6, H = 4, Rm =2, θ = −1.2 and different values of α.
−1 −0.5 0 0.5 1
θ
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
hx
N = 0.2H = 5m = 0.01
N = 0.8H = 10m = 40
Figure 10: Axial-induced magnetic field versus flow rate for α = 0.5 for different values of N at m =2, H = 4, Rm = 2 for different values of m at N = 0.7, H = 7, Rm = 1, and for different values of H atm = 4, N = 0.6, Rm = 2.
the Newtonian fluid lies below the curves for micropolar fluid in the pumping region, and inthe copumping region, the pumping decreases with an increase in N.
It is known that the stress tensor is not symmetric in micropolar fluid, that is why theexpressions for τxy and τyx are different. In Figures 4 and 5, we have plotted the shear stressesτxy and τyx at the upper and lower walls for various values of the Hartmann number H. Itcan be seen that both shear stress are symmetric about the line x = 0. However, its magnitudeincreases as H increases. Moreover, both shear stresses have directions opposite to the upperwave velocity, while the directions of these shear stresses are along the direction of the lowerwave velocity. Figure 6 indicates that the shear stress τyx decreases with an increase in themicrorotation parameter m, while it increases as the coupling number N increases, and so,
16 Journal of Applied Mathematics
−2 −1 0 1
Jz
−1.5
−1
−0.5
0
0.5
1
1.5
y
H = 2H = 4H = 6
H = 8H = 10
Figure 11: Variation of the current density distribution across the channel for α = 0.5, θ = 1.6, m = 3, N =0.2, Rm = 1, and different values of H.
−1 −0.5 0 0.5
Jz
−1.5
−1
−0.5
0
0.5
1
1.5
y
m = 3, N = 0.8m = 6m = 9Newtonian
N = 0.1, m = 3N = 0.3N = 0.5
Figure 12: Variation of the current density distribution across the channel for α = 0.5, θ = 1.6, H = 2, Rm =1, and different values of m and N.
the magnitude value of shear stress for a Newtonian fluid is less than that for a micropolarfluid.
4.2. Magnetic field characteristics
The variations of the axial-induced magnetic field hx across the channel at x = 0 and thecurrent density distribution Jz across the channel for various values of m, N, H, Rm, and αare displayed in Figures 7–12.
In Figures 7-8, m (= 5, 20, 100 with N = 0.9, Rm = 1, α = 0.4 and θ = 1.2 and differentvalues of H), N (= 0.2, 0.4, 0.6 with m = 2, Rm = 1, α = 0.4 and θ = 1.2 and different valuesof H).
Kh. S. Mekheimer 17
−4 −3 −2 −1 0 1 2
Jz
−1.5
−1
−0.5
0
0.5
1
1.5
y
Rm = 1Rm = 2Rm = 3
Figure 13: Variation of the current density distribution across the channel for α = 0.5, θ = 1.6, H = 4, m =3, N = 0.2, and different values of Rm.
These figures indicate that the magnitude of the axial-induced magnetic field hxdecreases as the microrotation parameter m and the Hartmann number H increases whileit increases as the coupling number N increases. Further in the half region, the inducedmagnetic field is one direction, and in the other half, it is in the opposite direction and it is zeroat y = 0. Figure 9 illustrates the variation of hx across the channel for different values of theamplitude ratio α at x = 0.5, where α (= 0, 0.3, 0.6, 0.9 with m = 3, N = 0.6, H = 4, Rm = 2,and θ = −1.2). It is clear that the magnitude of the axial-induced magnetic field hx at α = 0(no peristalsis) is larger, and it decreases with increasing α. The distributions of hx within thetime-averaged mean flow rate θ are exhibited in Figure 10 at x = 0.25, y = 0.5, where hx isplotted for various values of the parameters N (= 0.2, 0.8 with m = 2, H = 4, Rm = 2, α = 0.5), m (= 0.01, 40 with N = 0.7, H = 7, Rm = 1, α = 0.5) and H (= 5, 10 with m = 4, N =0.6, Rm = 2, α = 0.5). From this figure, we observe that there is an inversely linear relationbetween hx and θ for any value of the above-mentioned parameters, that is, hx decreaseswith increasing the flow rate θ and that the obtained curves will intersect at the point θ = 0.It is found also that for any value of θ ≤ 0, the effect of increasing each of M and H is todecrease hx values whereas the effect of increasing N is to increase the value of hx. On theother hand, for any value of θ ≥ 0, all the obtained lines will behave in an opposite mannerto this behavior when θ ≤ 0.
Figures 11–13 describe the distribution of the current density Jz within y for differentvalues of m, N, H, and Rm at the central line of the channel (x = 0), where H (= 2, 4, 6, 8,10 with m = 3, N = 0.2, Rm = 1, α = 0.5, θ = 1.6), m (= 3, 6, 9 with N = 0.8, H = 2, Rm =1, α = 0.5 and θ = 1.6), N (= 0.1, 0.3, 0.5 with m = 3, H = 2, Rm = 1, α = 0.5 and θ = 1.6),andRm (= 1, 2, 3, withm = 3, N = 0.2, H = 4, α = 0.5, θ = 1.6). The graphical results of thesefigures indicate that the dimensionless current density Jz decreases as H and M increase inthe region near the center of the channel while it increases for the same values of H and M inthe region near to the lower and upper walls, that is, the net current flow through the channelis zero and this corresponds to the case of open circuit. However, an opposite behavior isnoticed as N and Rm increases. Also, the current value for a micropolar fluid is higher thanthat for a Newtonian fluid.
18 Journal of Applied Mathematics
-0.05
-0.05
0.05
0.050.06
0.030.060.04
0
0
0
0.04
0.04-0.05
0.03
0.03
m = 3
(a)
-0.05
-0.05
0.05
0.06 0.04
0
0
0
0.05
0.04-0.05
0.04
m = 6
(b)
-0.05
-0.05
0.02
0.03
0.04
0
0
0
0.04 0.04
0.04 0.02
0.02
-0.05
0.03
0.03
−1.5 −1 −0.5 0 0.5 1 1.5
x
0
0.5
1
1.5
y
m = 9
(c)
Figure 14: Stream lines for different values of m.
4.3. Trapping phenomena and magnetic-force lines
Another interesting phenomenon in peristaltic motion is trapping. In the wave frame,streamlines under certain conditions split to trap a bolus which moves as a whole with thespeed of the wave. To see the effects of microrotation parameter m and the coupling numberN on the trapping, we prepared Figures 14 and 15 for various values of the parameters m(= 3, 6, 9 with N = 0.5, H = 2, θ = 0.7, α = 0.5) and N = (0.2, 0.4, 0.8 with m = 7, H =2, θ = 0.7, α = 0.5). Figures 14 and 15 reveal that the trapping is about the center lineand the trapped bolus decrease in size as the microrotation parameter m increases, whilethe size of the bolus increases as N increases. Also the effects of m and N on the magnetic-force function φ are illustrated in Figures 16 and 17 for various values of the parameters
Kh. S. Mekheimer 19
-0.03
0.02
-0.080.02
-0.03
-0.03
0.02
-0.08
-0.08
N = 0.2
(a)
0.04
-0.03
0.03 0.02
-0.08
0.04
0.02
-0.03
-0.03
0.02
0.03
-0.08
-0.08
0.030.03
N = 0.4
(b)
0.050.04
-0.030.02
-0.080.040.02
-0.03
0.02 -0.03
-0.08
-0.08
0.07
0.07 0.07 0.060.08
0.08
0.06
0.060.05
0.050.04
−1.5 −1 −0.5 0 0.5 1 1.5
x
0
0.5
1
1.5
y
N = 0.8
(c)
Figure 15: Stream lines for different values of N.
m (= 0.01, 3, 10, with N = 0.5, H = 2, θ = 0.7, Rm = 1, α = 0.4) and N = (0.1, 0.4, 0.8, withm = 3, H = 2, θ = 0.7, Rm = 1, α = 0.4). It is observed that as m increases, the size of themagnetic-force lines will decrease, and for large values of m, these lines will vanish at thecenter of the channel. An opposite behavior will occur as the coupling number N increase,where the width of the magnetic-force lines will increase and more magnetic-force lines willbe created at the center of the channel as the coupling number N increases.
5. Concluding remarks
The effect of the induced magnetic field on peristaltic flow of a magneto-micropolarfluid is studied. The exact expressions for stream function, magnetic-force function, axial
20 Journal of Applied Mathematics
0.05
0.1
0.10.1
0.1
0.1
0.05
0.05
0.05 0.150.15
0.15
0.19
0.19
0.2
0.2 0.21
m = 0.01
(a)
0.05
0.1
0.1
0.1
0.1
0.1
0.05
0.05
0.05
0.150.15
0.15
0.19
0.19
0.2
0.2 0.21
m = 3
(b)
0.05
0.1
0.10.1
0.1
0.1
0.05
0.05
0.05
0.150.15
0.15 0.18
0.19
0.19 0.2
0.18
−1.5 −1 −0.5 0 0.5 1 1.5
x
0
0.5
1
1.5
y
m = 10
(c)
Figure 16: Magnetic-force lines for different values of m.
pressure gradient, axial-induced magnetic field, and current density are obtained analytically.Graphical results are presented for the pressure rise per wave length, shear stresses on thelower and upper walls, axial-induced magnetic field, and current density and trapping. Themain findings can be summarized as follows.
(1) The value of the axial pressure gradient dp/dx is higher for a magneto-micropolarfluid than that for a Newtonian fluid.
(2) The magnitude of pressure rise per wavelength for a magneto-micropolar fluidis greater than that of a Newtonian fluid in the pumping region, while in thecopumping region, the situation is reversed.
Kh. S. Mekheimer 21
0.05
0.05
0.1
0.1
0.10.1
0.1
0.1
0.05
0.05
0.05
0.150.15
0.15 0.
17
0.17
0.16
0.16
0.16
N = 0.1
(a)
0.05
0.05
0.1
0.10.1
0.1
0.1
0.05
0.05
0.05
0.150.15
0.15
0.17
0.17
0.19
0.16
0.16
0.18
0.18
0.16
N = 0.4
(b)
0.05
0.1
0.1
0.1
0.1
0.1
0.05
0.05
0.05 0.150.15
0.15
0.19
0.19
0.2
0.2
0.21
0.21
0.22
−1.5 −1 −0.5 0 0.5 1 1.5
x
0
0.5
1
1.5
y
N = 0.8
(c)
Figure 17: Magnetic-force lines for different values of N.
(3) Shear stresses at the lower wall are quite similar to those for the upper wall exceptthat at the upper wall has its direction opposite to the upper wave velocity while atthe lower wall has its direction along the wave velocity.
(4) The shear stresses decrease with an increase of m and increases with increasing N.
(5) The axial-induced magnetic field is higher for a Newtonian fluid than that for amicropolar fluid and smaller as the transverse magnetic field increases.
(6) There is an inversely linear relation between the axial-induced magnetic field andthe flow rate.
22 Journal of Applied Mathematics
(7) The current density Jz at the center of the channel is higher for a micropolar fluidthan that for a Newtonian fluid, and it will decrease as the microrotation parameterand transverse magnetic field increases, while it increases as the coupling numberincreases.
(8) As we move from the Newtonian fluid to a micropolar fluid, more trapped boluscreated at the center line and the size of these bolus increases.
(9) More magnetic force lines created at the center of the channel as the fluid movesfrom Newtonian fluid to a micropolar fluid.
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